Lesson 28a - Approximating Taylor Series

Lesson 28a - Approximating Taylor Series - (x T f,a x n 0 f...

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Estimating Taylor Series C id T l l i l f f ti f( ) Consider a Taylor polynomial of a function f(x): ) ( , ! ) )( ( ) ( n n a f n a x a f x T Then we find that the approximation stopping after n = k will be given by: (somewhat similar to that of the alternating series test) 0 n )! 1 ( ) )( ( ) ( 1 ) 1 ( k a x z f x R k k k Where z is some (not usually known) number between a and x depending on how d i ti f d I h t th t R ( ) i th i d f many derivatives were performed. In here, we note that R k (x) is the remainder of how close f(x) is to the Taylor expansion stopping after k terms. Proving this theorem is a bit challenging, so we will not do so, but if we can place a reasonable bound on z, we can show that the error is
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Estimating Taylor Series S h d f( ) T( )? C id th f ll i Th So when does f(x) = T(x)? Consider the following Theorem: Consider a Taylor Series of a function f(x): 0 ) ( , ! ) )( ( ) ( n n n a f n a x a f x T Then T(x) = f(x) if and only if T(x) – f(x) = 0, if and only if Then T(x) f(x) if and only if T(x) f(x) 0, if and only if 0 ) ( ) ( lim x T x f k k 0 ) ( lim x R k k 0 )! ( ) )( ( lim 1 ) 1 ( k a x z f k k 1 k
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